Name: finding distance on a coordinate plane
Uploaded: 2025-11-26T05:44:23.000Z
Duration: 1 min 20 s
Channel: JoVE

Zwei Freunde betreten einen Park von unterschiedlichen Eingängen aus und bleiben an verschiedenen Ecken eines Feldes stehen.Von ihrer jeweiligen Position aus wollen sie den genauen Abstand zwischen ihnen bestimmen.Die Entfernungsformel wird angewendet, um den Abstand zwischen den beiden Positionen, die als Punkte A und B markiert sind, anhand ihrer Koordinaten zu berechnen.Es findet den kürzesten Weg zwischen ihnen – wie diagonal über ein Feld zu schneiden, anstatt um die Ränder herum zu gehen.Diese Berechnung basiert auf dem Satz des Pythagoras, wonach sich die horizontalen und vertikalen Abstände am Punkt C treffen und ein rechtwinkliges Dreieck bilden.Der Satz besagt, dass das Quadrat der Hypotenuse oder der diagonalen Seite gleich der Summe der Quadrate der beiden anderen Seiten ist.In einem Koordinatengitter stellen diese Seiten die Differenzen in den X- und Y-Werten zwischen den Punkten A und B dar.Indem man die horizontalen und vertikalen Differenzen quadriert, addiert und die Quadratwurzel zieht, wird die Länge der Hypotenuse bestimmt.Diese Beziehung ergibt die geradlinige Distanz zwischen den beiden Freunden.Solche Berechnungen werden auch bei der Kartierung von Routen und der Planung der Abdeckung drahtloser Netzwerke verwendet.

die abstands- und mittelpunktformel

In der Geometrie ist die Messung der euklidischen Entfernung zwischen zwei Punkten in der Ebene f&#252;r verschiedene praktische und theoretische Anwendungen von zentraler Bedeutung. Ob in der Navigation, im Ingenieurwesen oder in der Computergrafik &#8211; die Bestimmung des k&#252;rzesten Wegs zwischen zwei Orten beruht auf der Abstandsformel. Diese Formel leitet sich aus dem Satz des Pythagoras ab, der die L&#228;ngen der Seiten eines rechtwinkligen Dreiecks miteinander in Beziehung setzt. In einer Koordinatenebene dienen die horizontalen und vertikalen Abst&#228;nde zwischen zwei Punkten als Katheten eines rechtwinkligen Dreiecks, und die Verbindungslinie zwischen den Punkten bildet die Hypotenuse. Durch Berechnung der Quadratwurzel aus der Summe der Quadrate dieser horizontalen und vertikalen Differenzen l&#228;sst sich die euklidische Entfernung genau bestimmen.Ein Stadtplaner verortet zwei wichtige Standorte in einer Stadt: eine Bibliothek am Punkt L (300, 500) und einen Sportkomplex am Punkt S (1000, 1200). Zur Bestimmung der Distanz dieser Standorte wird die Abstandsformel verwendet:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:43.98%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40039/40039_Equation_1_1__editor_40039.svg' alt='40039_Equation_1_1_.svg' /></figure>Einsetzen der gegebenen Koordinaten:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:42.91%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40039/40039_Equation_2_1__editor_40039.svg' alt='40039_Equation_2_1_.svg' /></figure>Somit liegen die Bibliothek und der Sportkomplex etwa 990 m voneinander entfernt.Diese Methode veranschaulicht die praktische Anwendung geometrischer Prinzipien in der Stadtplanung und erm&#246;glicht eine pr&#228;zise Entfernungsmessung, die f&#252;r die Gestaltung effizienter Grundrisse und Infrastrukturen unerl&#228;sslich ist.

Die Abstands- und Mittelpunktformel

In der Geometrie ist die Messung der euklidischen Entfernung zwischen zwei Punkten in der Ebene f&#252;r verschiedene praktische und theoretische ...

Education

Lehre

JoVE Core

Math Fundamentals

Mathematische Grundlagen

Unit

Einheit

Coordinates and Graphs

Koordinaten und Graphen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

The Distance Formula

finding distance on a coordinate plane

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings.

The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them.

To find the distance, square the horizontal change and the vertical change, add those values, and take the square root. This gives the straight-line distance between the two points. The method is used in navigation, engineering, and computer graphics, where accurate distance matters.

A city planner example shows how the formula works in real life. A library is located at L (300, 500), and a sports complex is at S (1000, 1200). Using the distance formula, the two locations are about 990 meters apart. This kind of calculation helps planners design efficient layouts and infrastructure.

Finding Distance on a Coordinate Plane

coordinate plane

Finding Points on the Coordinate Plane

The Cartesian coordinate plane is used to find and plot points in two dimensions. It is made of two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. Their meeting point is the origin, where both values are zero. The axes divide the plane into four quadrants. These quadrants are labeled in a counterclockwise direction, starting from the upper right. Every point on the plane is written as an ordered pair. The first number is the...

the distance formula

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings. The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them. <br...

the midpoint formula

Finding the Center of a Line Segment

Finding the center of a line segment is a key idea in coordinate geometry. The midpoint is the point that lies exactly halfway between two points on a coordinate plane. It is used in mathematics, physics, engineering, and planning work where a central location matters. To work with a midpoint, start with two points, A(x1, y1) and B(x2, y2). A straight line segment can be drawn between them on the coordinate plane. The midpoint, labeled M, divides that segment into two equal parts. The...

graphs of equations in two variables

Plotting Two-Variable Equations with Tables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation. A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian...

circles

Circle Equations in Standard Form

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius. The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form. <br...

symmetry

Ellipse Symmetry at the Origin

An ellipse centered at the origin shows clear symmetry in its graph. Its equation describes a smooth, closed curve whose distance from the center keeps a constant ratio between the horizontal and vertical axes. Because the ellipse stretches farther along the x-axis than the y-axis, it has a horizontal shape. This ellipse has symmetry across the x-axis. That means the top half and bottom half are mirror images across the horizontal axis. Algebraically, the equation stays the same when y is...

solving equations graphically

Graphing Equations to Find Intercepts

Graphing equations helps students find intercepts and solutions on the coordinate plane. It gives a visual way to solve equations and understand how a function behaves. To use this method, write the equation in the form y = f(x). The solution to the original equation is the x-value where the graph crosses the x-axis. At that point, f(x) = 0. For example, the linear equation 2x - 4 = 0 can be written as y = 2x - 4. When this function is graphed, it has one x-intercept at x = 2, and that is the...

solving inequalities graphically

Graphing Inequality Solution Ranges

Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied. For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis. Inequalities can also compare two...

key terms and concepts

Learn how the distance formula uses the Pythagorean Theorem to measure straight-line distance between two points on a coordinate plane, with a city planning example.

Finding Distance on a Coordinate Plane | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Finding Distance on a Coordinate Plane

Description

Transcript

Tags